Kirszbraun theorem

In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H₁, and H₂ is another Hilbert space, and

${\displaystyle f:U\rightarrow H_{2}}$

is a Lipschitz-continuous map, then there is a Lipschitz-continuous map

${\displaystyle F:H_{1}\rightarrow H_{2}}$

that extends f and has the same Lipschitz constant as f.

Note that this result in particular applies to Euclidean spaces Eⁿ and E^m, and it was in this form that Kirszbraun originally formulated and proved the theorem.^[1] The version for Hilbert spaces can for example be found in (Schwartz 1969, p. 21).^[2] If H₁ is a separable space (in particular, if it is a Euclidean space) the result is true in Zermelo–Fraenkel set theory; for the fully general case, it appears to need some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient.^[3]

The proof of the theorem uses geometric features of Hilbert spaces; the corresponding statement for Banach spaces is not true in general, not even for finite-dimensional Banach spaces. It is for instance possible to construct counterexamples where the domain is a subset of ${\displaystyle \mathbb {R} ^{n}}$ with the maximum norm and ${\displaystyle \mathbb {R} ^{m}}$ carries the Euclidean norm.^[4] More generally, the theorem fails for ${\displaystyle \mathbb {R} ^{m}}$ equipped with any ${\displaystyle \ell _{p}}$ norm (${\displaystyle p\neq 2}$) (Schwartz 1969, p. 20).^[2]